发现公平PCA隐藏凸性，实现快速高效且公平的降维。
Hidden Convexity of Fair PCA and Fast Solver via Eigenvalue Optimization
- 通过特征值优化构建凸优化框架，突破公平PCA计算瓶颈。
- 真实数据集上比传统方法快8倍，仅慢于标准PCA不超过15%。
- 适合需要公平性保障的高维数据降维场景，如医疗与金融。
主成分分析（PCA）是机器学习中用于高维数据降维的基础技术，但可能产生对某些子群体不利的偏差结果。为解决此问题，Samadi等人（2018）提出公平PCA（FPCA）模型，旨在均衡不同子群体间的重构误差。然而，该模型基于半定松弛（SDR）的方法计算成本高昂，即使获得次优解也效率低下。为此，研究者提出了多种变体，但常偏离均衡重构误差的核心目标。本文揭示了FPCA模型中的隐藏凸性，并提出一种基于特征值优化的凸优化求解算法。该方法在不牺牲性能的前提下实现重构误差的公平性。实验表明，在真实数据集上，所提算法比基于SDR的方法快8倍，且仅比标准PCA慢最多85%。
原文摘要 · Abstract (English)
Principal Component Analysis (PCA) is a foundational technique in machine learning for dimensionality reduction of high-dimensional datasets. However, PCA could lead to biased outcomes that disadvantage certain subgroups of the underlying datasets. To address the bias issue, a Fair PCA (FPCA) model was introduced by Samadi et al. (2018) for equalizing the reconstruction loss between subgroups. The semidefinite relaxation (SDR) based approach proposed by Samadi et al. (2018) is computationally expensive even for suboptimal solutions. To improve efficiency, several alternative variants of the FPCA model have been developed. These variants often shift the focus away from equalizing the reconstruction loss. In this paper, we identify a hidden convexity in the FPCA model and introduce an algorithm for convex optimization via eigenvalue optimization. Our approach achieves the desired fairness in reconstruction loss without sacrificing performance. As demonstrated in real-world datasets, the proposed FPCA algorithm runs $8\times$ faster than the SDR-based algorithm, and only at most 85% slower than the standard PCA.
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